| L(s) = 1 | + 3-s + 5-s − 4.73·7-s + 9-s + 4.19·11-s − 1.26·13-s + 15-s − 6.92·17-s + 19-s − 4.73·21-s − 6·23-s + 25-s + 27-s − 10.1·29-s − 2·31-s + 4.19·33-s − 4.73·35-s + 2.73·37-s − 1.26·39-s + 11.6·41-s − 4.73·43-s + 45-s − 10·47-s + 15.3·49-s − 6.92·51-s + 4.19·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.78·7-s + 0.333·9-s + 1.26·11-s − 0.351·13-s + 0.258·15-s − 1.68·17-s + 0.229·19-s − 1.03·21-s − 1.25·23-s + 0.200·25-s + 0.192·27-s − 1.89·29-s − 0.359·31-s + 0.730·33-s − 0.799·35-s + 0.449·37-s − 0.203·39-s + 1.82·41-s − 0.721·43-s + 0.149·45-s − 1.45·47-s + 2.19·49-s − 0.970·51-s + 0.565·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.010016450796102564947735586508, −7.84327822722393718674513009035, −6.87969064854451314971609369608, −6.44341303902472256996622403150, −5.71505918988056270352534392382, −4.28502859918780940574269243520, −3.70681082271717832191169683704, −2.72802950633009112926375321664, −1.78812924739211803017917692003, 0,
1.78812924739211803017917692003, 2.72802950633009112926375321664, 3.70681082271717832191169683704, 4.28502859918780940574269243520, 5.71505918988056270352534392382, 6.44341303902472256996622403150, 6.87969064854451314971609369608, 7.84327822722393718674513009035, 9.010016450796102564947735586508
